#72314748

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#116356
commutator	C₂³×D₄	C₂²×C₄
Frattini, cosocle	C₂⁴×D₄	C₂³
prime radical	C₂²×#535	C₂²
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

2955 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,11,14,25)(4,8)(5,18,19,9)(6,10)(7,13)(12,22)(15,20,28,32)(16,30,31,23)(17,24)(21,29)(26,27);(1,3,5,13)(2,7,9,25)(4,15,16,12)(6,14,19,29)(8,22,23,32)(10,21,18,11)(17,28,31,27)(20,24,26,30);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,23)(10,24)(11,26)(13,28)(14,27)(15,29)(18,30)(19,31)(21,32)(22,25)

Polycyclic presentation:

2^10|a^2=e,b^2=df,b^a=bd,c^a=cf,c^b=cdeg,d^a=dh,d^b=di,d^c=dhi,e^b=eh,e^c=ehi,e^d=ej,f^a=fhi,f^b=fi,g^a=gj,g^b=gj,g^c=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314750, #72314760, #72314824, …
maximal normal subgroup of	#72314750, #72314760, #72314824, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4*F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);